The spectral degree exponent of a graph

TL;DR

The paper introduces the spectral degree exponent (SDE) as a novel graph metric for undirected, weighted, and possibly disconnected networks, linking it to degree assortativity through a rigorous analysis of its definition and extremal cases. The SDE q is defined by $\lambda_1 = \left( \frac{1}{N} \sum_{i=1}^N d_i^{\, q} \right)^{1/q}$, with proven existence and uniqueness and practical iterative methods (root finding, recursion) and bounds. It derives asymptotic expansions for several graph families (e.g., $P_N$, $W_N$, double-fork, modified lollipop) and demonstrates through simulations that the degree assortativity $\rho_D$ is the strongest correlate of the SDE across small graphs, random models, and real networks. The work shows that SDE is a robust, informative metric for assessing assortative mixing in graphs and provides a foundation for further theoretical and applied study, including potential applications in network design and analysis.

Abstract

We propose the spectral degree exponent as a novel graph metric. Although Hofmeister \cite{HofmeisterThesis} has studied the same metric, we generalise Hofmeister's work to weighted graphs. We provide efficient iterative formulas and bounds for the spectral degree exponent and provide highly accurate asymptotic expansions for the spectral degree exponent for several families of graphs. Furthermore, we uncover a close relation between the spectral degree exponent and the well-known degree assortativity, by showing high correlations between the two metrics in all small graphs, several random graph models and many real-world graphs.

Figures (9)

• Figure 1: The SDE $q$ is not monotonic in adding links, as exemplified in a star graph on 11 nodes. Sequentially adding non-existing links at random increases $q$ for the majority of the time, but not in all cases.
• Figure 2: The path graph $P_N$. In this example, $N=5$.
• Figure 3: The SDE $q$ for the path graph $P_N$ is computed by simulations (blue) and the approximate relation \ref{['eq_q_path']} (red).
• Figure 4: The wheel graph $W_N$. In this example, $N=7$.
• Figure 5: The "path with double fork graph" $A_N$ on $N+4$ nodes. In this example, $N=5$.

Theorems & Definitions (10)

• definition 1
• theorem 1: Existence and uniqueness HofmeisterThesis GS2
• definition 2
• theorem 2: Theorem 4.1 in hofmeister1988spectralradius
• theorem 3: Satz II.9 in HofmeisterThesis
• theorem 4
• theorem 5
• theorem 6
• theorem 7
• theorem 8